Is there a particular software package or language you're looking to solve this problem in? For someone not well versed in computational sciences, it might help with software recommendations.
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Geoff Oxberry♦Dec 28 '12 at 7:05

Any introductory text on numerical methods will have a chapter on this topic.
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David KetchesonDec 28 '12 at 20:02

What are $y_1$, $y_2$ and $y_3$ and how do they relate to $y(t)$?
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ja72Jan 4 '13 at 17:26

1 Answer
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You should be able to solve this problem using a multiple shooting method; you need only find initial conditions $y_{2}(0)$ and $y_{3}(0)$ that yield a solution consistent with your stated "final conditions". These values are typically called "boundary values"; your problem is called a two-point boundary value problem. It is worth noting that multiple shooting methods are more numerically stable than single shooting methods.

Thank you for the reply and for correcting my amateur wording of the question.
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user3489Dec 29 '12 at 6:08

No worries! I've had similar troubles getting up to speed in new fields. Sometimes, you just need a result to get a paper done, not a thorough grounding in a subject. As computational scientists, we should be helping people out with questions like yours, and I want to be a good ambassador for the community, because positive experiences are how we get people involved.
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Geoff Oxberry♦Dec 29 '12 at 7:23